Question

Three capacitances, each of $ 3\,\mu F $, are provided. These cannot be combined to provide the resultant capacitance of:
A) $ 1\,\mu F $
B) $ 2\,\mu F $
C) $ 4.5\,\mu F $
D) $ 6\,\mu F $

Answer 1

Hint
We can connect capacitors in series or parallel. Calculate the capacitance of all possible arrangements of 3 capacitors and check which option doesn’t match with the answers.
Formula used:
For capacitors in series: $ \dfrac{1}{{{C_{eq}}}} = \dfrac{1}{{{C_1}}} + \dfrac{1}{{{C_2}}} + \dfrac{1}{{{C_3}}}.... $
For capacitors in parallel: $ {C_{eq}} = {C_1} + {C_2} + {C_3}..... $

Complete step by step answer
To solve this question, we should calculate the net capacitance of all the possible arrangements of 3 capacitors which we are discussed below:
1.) 3 capacitors in series
Using the formula for the net capacitance of capacitors in series, we get
 $\Rightarrow \dfrac{1}{{{C_{eq}}}} = 3\left( {\dfrac{1}{3}} \right) = 1\,\mu F $
 $\Rightarrow {C_{eq}} = 1\,\mu F $
2.) 3 capacitors in parallel
Using the formula for the net capacitance of capacitors in parallel, we get
 $\Rightarrow {C_{eq}} = 3\left( 3 \right) = 9\,\mu F $
3.) 2 capacitors in series with one in parallel
For this combination, the net capacitance of the two capacitors in series will be
 $\Rightarrow \dfrac{1}{{{C_{||}}}} = 2\left( {\dfrac{1}{3}} \right) = \dfrac{2}{3} $
 $\Rightarrow {C_{||}} = \dfrac{3}{2} $
Then the net capacitance of all 3 capacitors will be
 $\Rightarrow {C_{eq}} = 3 + 1.5 = 4.5\,\mu F $
4.) 2 capacitors in parallel with one in series
For this combination, the net capacitance of the two capacitors in parallel will be
 $\Rightarrow {C_{series}} = 2\left( 3 \right) = 6\, $
Then the ned capacitance of all 3 capacitors will be
 $\Rightarrow \dfrac{1}{{{C_{eq}}}} = \dfrac{1}{6} + \dfrac{1}{3} = 0.5\, $
 $\Rightarrow {C_{eq}} = \,2\,\mu F $
Hence we cannot form a system that has a capacitance of option (D) - $ 6\,\mu F $.

Note
While the combination of three capacitors in series and parallel is very straightforward to guess, we should not forget the combinations in between where not all three are connected in series or parallel. While calculating the net capacitance, we should first evaluate the set of capacitors that are connected similarly so if 2 capacitors were connected in series with one in parallel, we should first calculate the capacitance of the two capacitors connected in series before calculating its net capacitance.